Let ^@ p ^@ be a prime number such that the next larger number is a perfect square. Find the sum of all such prime numbers. ^@(^@For example, if you think ^@ 11 ^@ and ^@ 13 ^@ are two such prime numbers, then the sum is ^@ 24.) ^@


Answer:

^@ 3 ^@

Step by Step Explanation:
  1. Given, ^@ p ^@ is a prime number such that the next larger number is a perfect square.
    ^@ i.e., p + 1 = n^2 ^@
  2. Rewriting ^@ p + 1 = n^2, ^@ we get
    ^@ \begin{align} & p = n^2 - 1 \\ \implies & p = (n - 1)(n +1) \end{align} ^@
  3. Since ^@ p ^@ is a prime number, therefore ^@ n-1 ^@ needs to be equal to ^@ 1. ^@
    ^@ \implies n = 2 ^@
    Therefore, ^@ p = 3 ^@ is unique.
    Hence, the required sum is ^@ 3. ^@

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